Solving $15x^2 - 2x - 8 = 0$ A Step-by-Step Guide

Hey everyone! Today, we're diving deep into solving the quadratic equation $15x^2 - 2x - 8 = 0$. Quadratic equations might seem daunting at first, but with the right approach, they become quite manageable. In this article, we’ll walk through the process step-by-step, ensuring you understand not just the how, but also the why behind each step. Let's get started!

Understanding Quadratic Equations

Before we jump into solving, let's quickly recap what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. The general form is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ represents the variable we're trying to solve for. The coefficients $a$, $b$, and $c$ play crucial roles in determining the nature and value of the roots (solutions) of the equation. For our equation, $15x^2 - 2x - 8 = 0$, we can identify $a = 15$, $b = -2$, and $c = -8$. Recognizing these coefficients is the first step in choosing the best method for solving the equation.

Why are quadratic equations so important? Well, they pop up everywhere in science, engineering, economics, and even everyday life. From calculating trajectories in physics to modeling growth patterns in biology, quadratic equations are indispensable tools. Mastering them opens doors to solving a wide array of real-world problems. In our specific case, we’ll be exploring different methods to find the values of $x$ that satisfy the equation. This involves understanding the structure of the equation and applying algebraic techniques to isolate $x$. Whether you're a student tackling algebra homework or someone brushing up on their math skills, this guide aims to provide a clear and thorough understanding of how to solve quadratic equations.

Methods to Solve Quadratic Equations

There are primarily three methods we can use to solve quadratic equations: factoring, completing the square, and the quadratic formula. Each method has its strengths and is suitable for different types of quadratic equations. Factoring is generally the quickest method when the quadratic expression can be easily factored. Completing the square is a more versatile method that can be used for any quadratic equation but is often more complex. The quadratic formula is a universal method that always works, regardless of whether the equation can be factored easily or not.

For the equation $15x^2 - 2x - 8 = 0$, we'll explore factoring first, as it’s often the most straightforward approach if applicable. Factoring involves breaking down the quadratic expression into the product of two binomials. If factoring proves difficult or impossible, we can then consider using the quadratic formula, which is a reliable backup method. Completing the square, while powerful, is generally more involved and might not be the most efficient choice for this particular equation. Understanding the nuances of each method allows you to choose the most appropriate one for the problem at hand. In this article, we'll primarily focus on factoring and using the quadratic formula to ensure a comprehensive understanding of how to tackle $15x^2 - 2x - 8 = 0$.

Factoring the Quadratic Equation

Factoring is a powerful technique for solving quadratic equations, especially when the coefficients allow for straightforward decomposition. The goal of factoring is to rewrite the quadratic expression $ax^2 + bx + c$ as a product of two binomials, like $(px + q)(rx + s)$, where $p$, $q$, $r$, and $s$ are constants. When we expand $(px + q)(rx + s)$, we should get back the original quadratic expression. This method relies on finding the right combination of constants that satisfy certain conditions.

For our equation, $15x^2 - 2x - 8 = 0$, we need to find two binomials that multiply to give us this expression. The first step is to look at the leading coefficient (15) and the constant term (-8). We need to find two numbers that multiply to the product of these two, which is $15 imes -8 = -120$, and add up to the middle coefficient, which is -2. This might sound tricky, but with practice, it becomes second nature. The numbers that satisfy these conditions are -12 and 10 because $-12 imes 10 = -120$ and $-12 + 10 = -2$. Now, we rewrite the middle term (-2x) using these numbers: $15x^2 - 12x + 10x - 8 = 0$. Next, we factor by grouping. We group the first two terms and the last two terms: $(15x^2 - 12x) + (10x - 8) = 0$. We factor out the greatest common factor (GCF) from each group: $3x(5x - 4) + 2(5x - 4) = 0$. Notice that we now have a common binomial factor, $(5x - 4)$. We factor this out to get $(5x - 4)(3x + 2) = 0$. Setting each factor equal to zero gives us the solutions. Let's move on to solving for $x$ in each factor.

Solving for x After Factoring

Once we've factored the quadratic equation, the next step is to find the values of $x$ that make the equation true. We do this by setting each factor equal to zero. This is based on the principle that if the product of two factors is zero, then at least one of the factors must be zero. In our case, we factored $15x^2 - 2x - 8$ into $(5x - 4)(3x + 2) = 0$. So, we have two equations to solve: $5x - 4 = 0$ and $3x + 2 = 0$.

Let's solve the first equation, $5x - 4 = 0$. To isolate $x$, we first add 4 to both sides of the equation: $5x = 4$. Then, we divide both sides by 5 to get x = rac{4}{5}. This is one of our solutions. Now, let's tackle the second equation, $3x + 2 = 0$. Similarly, we subtract 2 from both sides: $3x = -2$. Then, we divide both sides by 3 to get x = -rac{2}{3}. This is our second solution. Therefore, the solutions to the equation $15x^2 - 2x - 8 = 0$ are x = rac{4}{5} and x = -rac{2}{3}. These values of $x$ are also called the roots or zeros of the quadratic equation. They are the points where the parabola represented by the equation intersects the x-axis on a graph. Factoring allows us to find these roots directly by breaking down the quadratic expression into simpler, linear factors. But what if factoring isn't straightforward? That's where the quadratic formula comes to the rescue.

Using the Quadratic Formula

When factoring doesn't seem feasible or is proving too difficult, the quadratic formula is a reliable method to solve any quadratic equation in the form $ax^2 + bx + c = 0$. The quadratic formula is given by: $x = rac{-b ext{ ± } ext{√}(b^2 - 4ac)}{2a}$. This formula might look intimidating at first, but it's a powerful tool that guarantees a solution, regardless of the coefficients of the quadratic equation. The $ ext{±}$ symbol indicates that there are potentially two solutions: one where we add the square root term and one where we subtract it.

For our equation, $15x^2 - 2x - 8 = 0$, we have $a = 15$, $b = -2$, and $c = -8$. Plugging these values into the quadratic formula, we get: $x = rac-(-2) ext{ ± } ext{√}((-2)^2 - 4 imes 15 imes -8)}{2 imes 15}$. Simplifying this expression step-by-step, we first address the terms inside the square root $(-2)^2 = 4$ and $4 imes 15 imes -8 = -480$. So, the expression inside the square root becomes $4 - (-480) = 4 + 480 = 484$. The square root of 484 is 22. Now, we have: $x = rac{2 ext{ ± 22}{30}$. This gives us two potential solutions. For the addition case, x = rac{2 + 22}{30} = rac{24}{30} = rac{4}{5}. For the subtraction case, x = rac{2 - 22}{30} = rac{-20}{30} = -rac{2}{3}. These are the same solutions we found by factoring, which confirms the correctness of both methods. The quadratic formula provides a systematic way to find the roots of any quadratic equation, making it an indispensable tool in algebra. Let's delve into why this formula works and the discriminant's role in determining the nature of the solutions.

Understanding the Discriminant

The discriminant is a crucial part of the quadratic formula that gives us insights into the nature of the solutions (roots) of a quadratic equation. It's the expression under the square root in the quadratic formula: $b^2 - 4ac$. The value of the discriminant can tell us whether the quadratic equation has two distinct real roots, one real root (a repeated root), or no real roots (complex roots). This information is incredibly valuable because it helps us understand the behavior of the quadratic function and its graph.

Let's analyze the three possible scenarios:

1. If $b^2 - 4ac > 0$, the quadratic equation has two distinct real roots. This means the parabola intersects the x-axis at two different points. In our example, $15x^2 - 2x - 8 = 0$, the discriminant was $(-2)^2 - 4 imes 15 imes -8 = 484$, which is greater than 0. This confirms that we have two distinct real roots, which we found to be x = rac{4}{5} and x = -rac{2}{3}.
2. If $b^2 - 4ac = 0$, the quadratic equation has one real root (a repeated root). This means the parabola touches the x-axis at exactly one point. The root is repeated because both solutions from the quadratic formula will be the same.
3. If $b^2 - 4ac < 0$, the quadratic equation has no real roots. Instead, it has two complex roots. This means the parabola does not intersect the x-axis. The solutions involve imaginary numbers since we're taking the square root of a negative number.

Understanding the discriminant allows us to quickly assess the type of solutions we should expect before diving into the full solution process. It’s a powerful analytical tool that enhances our understanding of quadratic equations. In our case, the positive discriminant confirmed that we were on the right track to finding two real solutions. Now that we’ve covered the discriminant and the main methods for solving quadratic equations, let’s summarize our findings.

Conclusion

In this comprehensive guide, we've walked through the process of solving the quadratic equation $15x^2 - 2x - 8 = 0$. We explored two primary methods: factoring and the quadratic formula. Factoring allowed us to break down the quadratic expression into simpler factors, leading us to the solutions x = rac{4}{5} and x = -rac{2}{3}. When factoring wasn’t immediately obvious, we turned to the quadratic formula, a universal tool that guarantees a solution for any quadratic equation. By plugging in the coefficients $a = 15$, $b = -2$, and $c = -8$ into the formula, we arrived at the same solutions, reinforcing the correctness of both methods.

We also discussed the discriminant, $b^2 - 4ac$, which provides valuable insights into the nature of the roots. A positive discriminant, as we had in our case (484), indicates two distinct real roots. Understanding the discriminant helps us anticipate the type of solutions we should expect, making the solving process more efficient.

Mastering quadratic equations is a fundamental skill in algebra and has wide-ranging applications in various fields. By understanding the different methods and tools available, such as factoring and the quadratic formula, you’ll be well-equipped to tackle a variety of problems. Whether you're a student looking to ace your math exams or someone wanting to brush up on their algebra skills, the ability to solve quadratic equations is a valuable asset. Keep practicing, and you'll find that these equations become less daunting and more manageable. Thanks for joining us on this mathematical journey!